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Monte Carlo-based calibration and uncertainty analysis of acoupled plant growth and hydrological model

T. Houska, S. Multsch, P. Kraft, H.-G. Frede, and L. Breuer

Institute for Landscape Ecology and Resources Management, Research Centre for BioSystems, Land Use and Nutrition (IFZ),Justus Liebig University Giessen, Giessen, Germany

Correspondence to:T. Houska ([email protected])

Received: 14 October 2013 – Published in Biogeosciences Discuss.: 11 December 2013Revised: 13 February 2014 – Accepted: 27 February 2014 – Published: 11 April 2014

Abstract. Computer simulations are widely used to supportdecision making and planning in the agriculture sector. Onthe one hand, many plant growth models use simplified hy-drological processes and structures – for example, by the useof a small number of soil layers or by the application of sim-ple water flow approaches. On the other hand, in many hy-drological models plant growth processes are poorly repre-sented. Hence, fully coupled models with a high degree ofprocess representation would allow for a more detailed anal-ysis of the dynamic behaviour of the soil–plant interface.

We coupled two of such high-process-oriented indepen-dent models and calibrated both models simultaneously.The catchment modelling framework (CMF) simulated soilhydrology based on the Richards equation and the vanGenuchten–Mualem model of the soil hydraulic properties.CMF was coupled with the plant growth modelling frame-work (PMF), which predicts plant growth on the basis of ra-diation use efficiency, degree days, water shortage and dy-namic root biomass allocation.

The Monte Carlo-based generalized likelihood uncertaintyestimation (GLUE) method was applied to parameterize thecoupled model and to investigate the related uncertainty ofmodel predictions. Overall, 19 model parameters (4 for CMFand 15 for PMF) were analysed through 2× 106 model runsrandomly drawn from a uniform distribution.

The model was applied to three sites with different man-agement in Müncheberg (Germany) for the simulation ofwinter wheat (Triticum aestivum L.) in a cross-validation ex-periment. Field observations for model evaluation includedsoil water content and the dry matter of roots, storages, stemsand leaves. The shape parameter of the retention curven washighly constrained, whereas other parameters of the retention

curve showed a large equifinality. We attribute this slightlypoorer model performance to missing leaf senescence, whichis currently not implemented in PMF. The most constrainedparameters for the plant growth model were theradiation-useefficiencyand thebase temperature. Cross validation helpedto identify deficits in the model structure, pointing out theneed for including agricultural management options in thecoupled model.

1 Introduction

Plant growth and hydrological models are widely used toevaluate strategies such as climate adaption, risk manage-ment of pesticide, or fertilizer application in agricultural sci-ences and politics. However modelling comes along withlimitations. Different models can lead to deviating resultseven if they are driven by the same input and forcing data.Such effects are represented by model uncertainty. Further-more, the selection of input parameters can change the re-sults and also increase uncertainty. This effect is commonlyknown as parameter uncertainty. Hence, a good knowledge ofthese uncertainties is crucial, especially when plant growthmodels are used to project food supply and hydrologicalmodels are applied in order to develop strategies for wa-ter resource management. The importance of a comprehen-sive knowledge about the capabilities and limitations of suchmodels applied in the field of decision making has also beenhighlighted by Kersebaum et al. (2007).

Most current plant growth models integrate plant growthand hydrological processes tightly, leading to very complexmodels. Therefore, the calibration of such models is often

Published by Copernicus Publications on behalf of the European Geosciences Union.

2070 T. Houska et al.: Monte Carlo-based calibration and uncertainty analysis

done step by step. In a number of studies (e.g. Pathak et al.,2012; Wang et al., 2005; Iizumi et al., 2009) the hydrolog-ical model has been calibrated as a first step and the plantgrowth model as a second step in order to reduce the num-ber of parameters varied in one calibration step. However,in such a setup, feedbacks between biomass production andhydrology are not considered (Pauwels et al., 2007). Alterna-tively, the past years have seen modular model developmentsand the promotion of comprehensive model-coupling strate-gies (Priesack et al., 2006). Kraft et al. (2011) coupled thecatchment modelling framework (CMF) (Kraft et al., 2011)with the plant growth modelling framework (PMF) (Multschet al., 2011) to simulate the direct feedbacks of soil hydrolog-ical conditions on plant development. However, their coupledversion of CMF and PMF has so far only been used for vir-tual modelling experiments (Multsch et al., 2011, Kraft et al.,2011) and not yet for real observed data.

Instead of calibrating single models step by step, we favourthe use of a Monte Carlo algorithm to screen the hyper-dimensional parameter space for behavioural model runs ofthe entire coupled model and apply the GLUE (general-ized likelihood uncertainty estimation) method, a widespreadBayesian technique to investigate model performance andparameter uncertainty (Beven and Binley, 1992). GLUE re-sults in a range of parameter sets which all lead to accept-able model runs rather than only one “optimal” calibratedparameter set (Candela et al., 2005). This behaviour is knownas “equifinality”. Model realizations are grouped into be-havioural and non-behavioural model runs and associatedparameter sets. The former describes an acceptable modelapplication, allowing for some degree of error in simulat-ing a target value (defined in an a priori threshold crite-rion). The latter describes parameter sets which return un-acceptable model outputs and can be deleted (Beven, 2006).A further distinction is made between constrained and un-constrained parameters (Christiaens and Feyen, 2002). Themore sensitive a model parameter for predicting a given tar-get value is, the more constrained it becomes in the remainingbehavioural parameter sets.

The level of improvement of the model by the GLUE ap-proach depends on the used likelihood function threshold cri-terion and the number of sampled parameters. However, thechoice of the likelihood function itself also has a strong in-fluence on the results, which has also been reported by Heet al. (2010), who additionally highlight the importance ofthe likelihood function to ensure statistical validity. A num-ber of likelihood functions have been applied: for example,the inverse error variance with a shaping factor (Beven andBinley, 1992), the Nash and Sutcliffe model efficiency (Freeret al., 1996), scaled maximum absolute residuals (Keesmanand van Straten, 1990) and the index of agreement (Wilmott,1981), model bias and coefficient of determination.

A number of studies applied the GLUE method to achievea better understanding of plant growth models and theirparameters. For example, Wang et al. (2005) utilized the

GLUE method for evaluation of the EPIC model with themean squared error as a likelihood function. He et al. (2010)tested the influence of different likelihood functions withthe crop environment resource synthesis (CERES)-Maizemodel. They used modifications of the variance of modelerrors and mean squared error as likelihood functions. Moand Beven (2004) applied the method with the index ofagreement as a likelihood function for calibration of a soil–vegetation–atmosphere transfer model. Pathak et al. (2012)considered bias, root-mean-squared error (RMSE) and theindex of agreement as likelihood functions in the uncertaintyassessment of the CSM-CROPGRO-Cotton model.

In this study, the combined set of model parameters from afully coupled plant growth model (PMF) and a hydrologicalmodel (CMF) was calibrated parallelly. Hence, the objectivesof this study were as follows:

– In-depth analysis of the coupled model setup througha GLUE analysis to investigate the sensitivity of plantgrowth and hydrological model parameters and to de-rive a range of behavioural model runs.

– Cross validation of parameter transferability for threedifferent sites against observed soil moisture andbiomass data for storages and roots as well as stemsand leaves (further summarized as plant dry matter) ofwinter wheat.

To describe the “goodness of fit” of our model prediction,we used a set of three likelihood functions (model efficiency,bias and coefficient of determination). Subsequently, we willdistinguish between (i) forcing data (e.g. meteorological ob-servations), (ii) input data (e.g. soil information) and (iii)model parameters (e.g. plant coefficients).

2 Materials and methods

2.1 Model setup

2.1.1 Catchment modelling framework (CMF)

A plot-scale hydrological model for the unsaturated zone wasbuilt by using the catchment modelling framework (CMF)(Kraft, 2011). CMF is a computer program used for settingup individual hydrological models. A programming libraryfacilitates the design of water transport models between soillayers in up to three dimensions. It allows the developmentof detailed mechanistic models as well as lumped large-scalelinear storage-based models. A model in CMF functions asa network of storages and boundary conditions connectedby flux-calculating sub-models. It works as an extension toPython and can easily be coupled with other models.

The specific realization of CMF was done with a one-dimensional setup. Water fluxes were simulated with theRichards equation. We simulated the soil moisture with the

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T. Houska et al.: Monte Carlo-based calibration and uncertainty analysis 2071

van Genuchten–Mualem model of the soil hydraulic proper-ties (van Genuchten, 1980) for 50 soil layers with a uniformthickness of 0.05 m. Theksat parameter was used to simulatethe saturated conductivity. The porosity parameter is definedby pore volume per soil volume, while alpha andn as knownvan Genuchten parameters. The interaction of the lowest soillayer with the groundwater is modelled as a Neumann bound-ary condition. To initiate the water content of CMF we usedmeteorological data for the year 1992 and calibrated it for thegrowing season 1993/1994.

2.1.2 Plant growth modelling framework (PMF)

As a plant growth model, we used the plant growth mod-elling framework (PMF), developed by Multsch et al. (2011).PMF is a dynamic and integrative tool for setting up individ-ual plant models. In general, PMF consists of four core ele-ments: (i) plant model, (ii) process library, (iii) plant buildingset and (iv) crop database. The basic idea of PMF is to di-vide the plant into its parts – root, shoot, stem, leaf and stor-age – which interact during the growth process. This struc-ture builds up the plant model. A process library containsmathematical formulations of biophysical processes, such asbiomass accumulation, water uptake and development. Theuser can connect the plant model with a set of biophysicalprocesses by using the plant building set. The plant parame-ters are taken from the crop database.

The biomass accumulation is affected by the radiation useefficiency (RUE). The higher the RUE, the higher is thebiomass accumulation. RUE is used to calculate the biomassgrowth with the biomass radiation-use-efficiency concept(Monteith and Moss, 1977). The mathematical solution ofthe radiation use efficiency in PMF is based on Acevedoet al. (2002). The photosynthetically active absorbed radia-tion is calculated by solar radiation and its intercepted frac-tion. The simulation of biomass accumulation from photo-synthetic active radiation is performed with the canopy ex-tinction coefficient (k), where the radiation is directly trans-formed into dry matter or assimilated CO2. An overview ofthis concept is given by Sinclair and Horie (1989). The min-imum temperature for plant development is defined by thebase temperature (tbase). This acts as a threshold tempera-ture above which development occurs. Each plant develop-ment step is defined by a temperature sum. If the tempera-ture sum is reached, then the developing process begins. Ifthis parameter is too high, then the plant starts its growingprocess too late, and vice versa. For simplicity, the parametertbase is independent from further environmental influences.Development stages are used to control biomass allocationbetween plant organs. The plant development is divided intothe eight stages by a thermal time threshold: emergence, leafdevelopment, tillering, stem elongation, anthesis, seed fill,dough stage and maturity. Root elongation determines thedaily root growth rate. Root water uptake in PMF equals amacroscopic approach type 2 (Feddes et al., 2001; Hopmans

and Bristow, 2002). In this concept the actual water uptakeis distributed over the rooting zone and occurs as a sink termin the Richards equation. The allocation of water uptake inPMF depends on the relation of the root biomass in each soillayer and the total root biomass in the rooting zone. Influ-ences of water are incorporated according to the soil mois-ture conditions with a crop-specific response function. Theresponse function is related to the soil matrix potential andthe water content. According to Feddes et al. (1978), the re-sponse function is determined by three thresholds definingoxygen deficiency, wilting point and optimal water uptakeconditions. The crop-specific response function includes adimensionless water stress index. The resulting actual wa-ter uptake from each layer is the product of the stress indexand the available water.

The root growth takes place during sowing and the devel-opment stage anthesis. During this part of the growing sea-son, a fraction of the total biomass is allocated to the root.Root growth includes the calculation of the total undergroundbiomass as a fraction from the total plant biomass, the cal-culated vertical growth (elongation) and the distribution ofthe root biomass over the rooting zone (branching). The lastgroup of parameters are the basal crop coefficients (kcbini ,kcbmid and kcbend), which are used to assess plant transpira-tion from potential evapotranspiration (Table 1). The transpi-ration and evaporation in PMF are simulated according to thedual-coefficient approach described by Allen et al. (1998).The potential PET is adjusted with crop-specific coefficientsto account for different growing stages. This crop coefficientis low at the end of the growing period of winter wheat toaccount for a lower transpiration as a consequence of leafsenescence. For our case study, because of the lack of avail-able phenological data, we did not activate the vernalizationmodule. The important process of vernalization will be testedin future when phenological data are available. Plant parti-tioning is done according to biomass fractions of each plantorgan according to a table given by De Vries (1989). Thefraction of biomass which is allocated to each organ dependson the growing stage. Root growth and stem elongation oc-curs until anthesis. After that stage, dry matter is only al-located to the above-ground biomass. At the very end of thegrowing season the storage organs are filled. All PMF param-eters are chosen on the basis of their influence on roots, stemsand leaves or storages dry-matter outputs, based on one-parameter-at-a-time sensitivity analyses and expert knowl-edge.

2.1.3 Coupling CMF and PMF

Since both models contain Python interfaces which exposeall states and fluxes, the implementation of “glue” code toconnect the exchange of data is trivial. However, the defi-nition of the functional boundary between the models is amajor issue in model coupling – that is, which processesare covered by which model. It necessary to be ensure that

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Table 1. Parameter ranges of the Monte Carlo simulation for the coupled CMF–PMF for site 1 in Müncheberg. PAR stands for photosyn-thetically active radiation. Minimal to maximal input is the range for the GLUE analysis, while the output is the constrained range of theobserved behavioural parameter sets (cf. Figs. 2 and 3). Uncertainty reduction in the output over 30 % is reflected in bold type.

Parameter Definition Min. Max. Min. Max. Reductioninput input output output [%]

CMFalpha inverse of the air entry potential [cm−1] 0.001 0.1 0.007 0.1 6ksat saturated conductivity [m day−1] 0.1 25 4.1 24.9 16porosity pore volume per soil volume in [m3 m−3] 0.3 0.7 0.35 0.7 13n shape parameter of retention the curve [–] 1 2 1.3 1.7 60

PMFRUE radiation use efficiency [g MJ−1 PAR] 0 15 1.5 4.9 77k canopy extinction coefficient in [−] 0.2 0.8 0.2 0.79 2tbase min. temp. above growth can take place [◦C] −1 5 −0.2 3.8 33root growth root elongation factor in [cm day−1] 0.1 3 0.15 2.9 5emergence 144 176 144 174 6leaf development 176 229 176 228 2tillering 229 463 230 380 36stem elongation total thermal time requirement 463 725 466 724 2anthesis for each growing stage [◦ days] 725 991 748 989 9seed fill 991 1291 994 1284 3dough stage 1291 1672 1292 1669 1maturity 1672 1832 1683 1831 8kcbini 0.05 0.4 0.08 0.22 60kcbmid basal crop coefficient [−] 0.5 1.5 0.58 1.4 18kcbend 0.075 0.225 0.077 0.223 3

processes are not simulated by both models. In this study,the problem is solved by the definition of interfaces in eachmodel to transport information transparently.

The plant model contains methods for the connection tothe environmental interfaces (“soil”, “atmosphere”) and theclasses of the process library (“water”, “ET”). The methodsof the classes “soil” and “atmosphere” are implemented toquery the state variables and parameters of CMF directly.Hence, at any code block, where PMF needs informationabout the actual matrix potential, water content and meteo-rological conditions, this information is routed through theseclasses to CMF, which sends the data back.

The class “ET” calculates the evapotranspiration and theclass “water” estimates the plant’s water stress due to soilmoisture conditions. During every day in the growing season,the plant model calls the class ET for the calculation of thepotential transpiration (Tpot). A water stress value for eachlayer (ws) is calculated and contained inwstress. Finally, theactual transpiration is determined by the sum of the stress-driven water uptake from the rooting zone and the water up-take of the plant from each layer by the proportion of fine-root content and water stress of each layer.

Each layer of the water transport model has an accessi-ble Neumann boundary condition representing the systemboundary between the soil and the roots. The flux of theboundary conditions is set by the coupling code to the wa-

ter uptake calculated by PMF. Using the interface approach,the governing equation of water transport, plant growth andpotential transpiration can be changed without changing thecoupling system. An extension from plots to larger scalessuch as a hillslope, as shown for a virtual case by Kraft etal. (2011), or a full 3-D landscape model is hence feasiblewithout the need to change PMF and the coupling mecha-nism.

The two models are run consecutively after each time stepusing an operator split approach. First the plant growth modelPMF simulates one time step,t , using the states of the watertransport model CMF att − 1. After that, CMF proceeds tot using the water uptake of PMF as a loss term at each soillayer.

2.2 GLUE setup

2.2.1 Likelihood functions

The performance of a parameter set to predict observationswas evaluated by goodness-of-fit values, represented by sev-eral likelihood functions. The choice of the likelihood func-tion depends on the situation and is often subjective (Bevenand Binley, 1992). Nevertheless, the choice of only one ob-jective function for the calibration is inaccurate in most cases(Vrugt et al., 2003). We therefore used a combination of threelikelihood functions:



1. The bias function was used as a central statistical mea-surement to summarize overall model performance:

L1 (θ |Y) = Bias=1

N

N∑i

Yi − Yi, (1)

where L(θj |Y

)is the likelihood measure for each

model run with parameter setθ , N is the total num-ber of measurements,Yi is the measured value for theith measurement andYi is the corresponding outputof the model. The bias measures the differences be-tween measurements and model outputs. For under-predictions of the model, the bias is positive; for over-predictions, the bias is negative. Thus, it is a usefulmeasure for assessing whether structural changes ofthe model equations are necessary for reducing theoverall bias of the prediction. (Wallach, 2006). How-ever, bias alone is not sufficient to evaluate model er-rors, as a bias of zero could also be due to cancellationof large errors with different signs (Wallach, 2006).

2. In order to measure the deviation of model predictionand measurement data we used the coefficient of de-termination, which is defined as

L2 (θ |Y) = R2=

N∑

i=1

[(Yi − Y

)(Yi −

¯Y

)](

N∑i=1

(Yi − Y

)2 N∑i=1

(Yi −

¯Y

)2)0.5

2

, (2)

whereY is the average of the measured and¯Y is the

average of the simulated data. A maximum value ofR2

= 1 indicates that a perfect linear relationship be-tween measured and calculated values exists, while theminimum value ofR2

= 0 indicates a low performanceof the model.R2 alone is also not a good measureof the model agreement with the observations, asR2

could also be equal to 1 if the model systematicallyover- or under-predicts.

3. Finally, we employed the Nash–Sutcliff index (Nashand Sutcliffe, 1970) for measuring the model’s sensi-tivity to outliers. This widely used function (e.g. Gar-nier et al., 2001; Beven and Binley 1992) is calculatedas follows:

L3 (θ |Y) = NSE= 1−

N∑i=1

(Yi − Yi

)2

N∑i=1

(Yi − Y

)2. (3)

If the model predicts the measurements perfectly, wehaveYi = Yi , implying NSE= 1. If Yi = Y for all i,then NSE= 0. Thus, a model which gives NSE= 0 hasthe same goodness of fit as using the average of themeasured data for every situation (Wallach, 2006).

The three proposed likelihood functions cover most as-pects in an adequate manner. They are widely used in hy-drology (e.g. Li et al., 2010; Besalatpour et al., 2012; Pathaket al., 2012) with high explanatory power. It has to be notedthat other choices for the likelihood function would certainlybe imaginable (Beven and Freer, 2001).

2.2.2 GLUE sequence

The general GLUE method proceeds in several consecutivesteps (Beven and Binley, 1992), which were adapted to thisspecific study:

a. Selection of sensitive parameters: a full list of all pa-rameters considered in GLUE from CMF and PMFis given in Table 1. Fifteen plant specific parametersfrom PMF which influence plant development, tran-spiration and biomass production were altered in theanalysis. The hydrological parameters were given bythe van Genuchten–Mualem parameters. These param-eters were selected on the basis of a one-parameter-at-a-time sensitivity analysis, which is not presented inthis study.

b. Creation of a priori distribution: a random functionwas used to create 2× 106 parameter sets, wherebyeach parameter set consisted of 19 parameters. Sincetheir a priori distribution was unknown, a uniform dis-tribution was assumed. The parameter ranges were se-lected on the basis of expert knowledge and other pub-lications.

c. Execution of model runs: the 2× 106 realizations ofthe coupled CMF–PMF model were forced with thesame climate data for three different sites by using ahigh-performance computing cluster.

d. Creation of posteriori distribution: the simulated vari-ables of both models were compared with observeddata by using the three likelihood functions NSE,R2

and bias. The variables of PMF are root, stem andleaves as well as storage dry matter and soil moisturein the case of CMF. Three threshold criteria were usedto obtain parameter settings which fit the measureddata equally well. All parameter sets that resulted ina bias> ±500 kg ha−1 for the plant dry matter and> ±10 % soil moisture, an NSE< 0 and aR2 < 0.3,were discarded.

These four steps resulted in behavioural parameter sets forthe coupled model for each study site. In order to test thelimitations of the behavioural parameter sets, a full cross val-idation for all three sites was conducted.

2.3 Study site and data

Study site: the coupled CMF–PMF model was parameterizedand evaluated using data from three agricultural field sites.



They are located at the Müncheberg experimental stations,50 km to the east of Berlin, Germany, where the ZALF (Leib-niz Centre for Agricultural Landscape Research) recorded acomprehensive experimental data set (Mirschel, 2007). Thisextensive data set was used in several previous modellingstudies (e.g. Wegehenkel, 2000; Palosuo et al., 2011; Kerse-baum et al., 2007). Sites are characterized by a primarilysandy Eutric Cambisol, with a homogenous volumetric sandcontent of 80 to 90 % in a soil profile with 2.25 m depth. Siltand clay content contribute 5 to 10 %. The soil is medium-textured with good structural stability. The bulk density isaround 1.5 g cm−1 and the organic matter content in the first0.3 m amounts to 0.6 %. An in depth description of soil phys-ical and chemical properties is given by Mirschel (2007).

Forcing data: the climate data comprise daily sum of pre-cipitation, minimum and maximum temperature, mean rela-tive humidity, early morning vapour pressure, global radia-tion and mean wind speed. During 1994 (the year for whichwinter wheat was cultivated), the conditions were relativelyhumid, with an annual precipitation of 714 mm and an an-nual average temperature of 9.1◦C. During the growing sea-son (October–July), precipitation of 588 mm and an averagetemperature of 15.8◦C were measured. The day of sowingand the day of harvest were set to observed dates (15 Octo-ber 1993 and 29 July 1994).

Evaluation data: average gravimetric soil moisture mea-surements are available for three soil depths (0–0.3, 0.3–0.6and 0.6–0.9 m) on 15 days during the observation periodfrom 1993 to 1994 for every site. The average soil mois-ture ranged from 12.1 to 12.9 % at site 1–3, with a minimumof 3 % and a maximum of 21.2 %. Soil moisture was verysimilar across all three sites. The sites differed in their man-agement strategies, with high-level intensive (site 1), organic(site 2) and extensive management (site 3) and in their win-ter wheat cultivar, namely ’Bussard’, ’Ramiro’, and ’Greif’.Crop growth data for winter wheat are available for fivedifferent days in 1994. Data on root dry matter [kg ha−1],stem and leaves dry matter [kg ha−1] and storage dry matter[kg ha−1] are given for all three sites.

3 Results and discussion

3.1 Parameter uncertainty

To assess the range of each parameter in the behavioural pa-rameter sets, we need to take a closer look at the parameterdistributions for the different likelihood functions. Table 1summarizes the results of the GLUE approach, providing thea priori and posteriori parameter ranges for the 19 modelparameters as well as the reduction of the parameter uncer-tainty. For five parameters we were able to substantially re-duce their uncertainty bounds by 30 to 70 %, while 11 param-eters were rather unconstrained, with uncertainty reductionof less than 10 %. Out of the eight parameters that define the

growing stage through the thermal time requirement [◦ days]only the parameter tillering showed a large uncertainty re-duction potential. This indicates that many of the parametersidentifying plant growth stages lead to a high grade of equi-finality.

A selection of eight model input parameters in terms of be-havioural model runs is shown in Fig. 1, where four CMF andfour PMF parameters are given as interaction scatter plots.The figure depicts the mean NSE (calculated from the sin-gle, equally weighted NSE for soil moisture, roots, stems andleaves, as well as storage dry matter) for site 1 as an exam-ple for constrained as well as non-constrained parameters ofTable 1. On the interaction scatter plots, no correlations be-tween parameters of PMF and CMF can be detected (Fig. 1).As the GLUE method cannot deal with such correlations, thisis an important precondition of the GLUE method (Jin et al.,2010). The interaction scatter plots also show a clear predic-tion boundary for the parametern at 1.3 [–] and for RUE at6 g MJ−1 PAR. A setting of RUE above 6 g MJ−1 PAR andn below 1.3 [–] can never lead to an adequate model pre-diction for winter wheat and soil moisture in 1994 for site 1in Müncheberg, regardless of which values are selected forother model input parameters.

The most constrained parameter of PMF is RUE (Ta-ble 1, Fig. 1), which influences biomass accumulation. Goodsettings for RUE were found from 1.5 to 4.9 g MJ−1 PAR(Table 1).This range is in line with most other applica-tions. Acevedo et al. (2002) suggested a RUE of 3.0 g MJ−1

PAR for wheat, which was used in the setup of Multschet al. (2011). Calderini et al. (1997) found RUE acrosstheir investigated wheat cultivars between 1.08 and 1.16 gMj-1 PAR. Lindquist et al. (2005) suggested a RUE of3.8 g MJ1 PAR. DSSAT version 4.0 uses RUE with a settingof 4.2 g MJ−1 PAR (Jones et al., 1998). Due to large influenceof weather variability to the RUE response, CERES-Maizedevelopers do not recommend using RUE as a calibration pa-rameter (Ma et al., 2011). We therefore suggest fixing RUEat our found optimum of 2.02 g MJ−1 PAR for further appli-cations of PMF.

The second-most-constrained parameter (Table 1, Fig. 1)is the CMF shape parameter of the retention curven, witha strict optimum at 1.45 [–] and a range reduction of 60 %through the threshold criteria. Christiaens and Feyen (2002)foundn to be not greatly constrained from 1.2 to 1.6 for theMIKE SHE model. In contrast, Vogel et al. (2000) reportedthe parameter to be quite sensitive. Ippisch et al. (2006)demonstrated that the van Genuchten–Mualem model causedconvergence problems withn close to 1.0 for the numericalsolver, which we can confirm.

When looking at the density distribution of the behaviouralmodel runs in Fig. 1, we can locate an optimal parameterrange with 0.015–0.025 [−] for alpha. But even within thisrange there is no guarantee of a good model response. Sev-eral parameter values in the considered range of alpha yieldpoor prediction with an NSE< 0, depending on the settings



Fig. 1. Parameter uncertainty and interaction. The scatter plots show parameter interaction and correlations for behavioural model runscoloured from yellow to red for NSE> 0 at site 1 in Müncheberg for the coupled CMF–PMF model. PMF parameters are given on thex

axis, and CMF parameters are plotted on they axis. The density distributions at the top and to the right depict the parameter uncertainty.NSE are reported as mean, equally weighted NSE for soil moisture, root, stem and leaves, as well as storage dry matter.

of other model parameters. The parametertbaseprovides bestresults within the range of 1.5 to 2.5◦C (Table 1). These set-tings are higher than the value given for PMF by Multschet al. (2011) with 0◦C, which was based on a study by Mc-Master and Wilhelm (1997). A wide range from 0 to 10◦Cfor tbase can be found in the literature, and is strongly de-pendent on cultivars (Porter and Gawith, 1999). The rootgrowth shows a local optimum at 0.6 and a global optimumat 2.4 cm day−1. Following our GLUE results thek parame-ter could not be confined; nevertheless, Pathak et al. (2012)found thek parameter constrained to around 0.64 [–] for theCROPGRO-Cotton model. All other investigated parametersare unconstrained within their boundaries to the outputs ofvarious plant dry matter and soil moisture (Table 1, Fig. 1).

Finding mainly insensitive parameters corresponds wellto prior studies using the GLUE procedure for models with

a similar large number of model parameters (e.g. Viola etal., 2009; Rankinen et al., 2006). Part of the problem is thatparameter-rich models allow for equifinality, levelling out theimpact of certain parameters.

3.2 Model performance

3.2.1 Soil water balance

Figure 2 summarizes the capability of the coupled CMF–PMF model for predicting the soil moisture output in threedepths. In addition to the median of the GLUE-derived be-havioural model runs, we also showed the 50 and 95 % uncer-tainty bounds. Overall, approximately 90 % of the observeddata were within the predicted uncertainty bounds. The un-certainty of the prediction was higher during dry and wet



27

1

2

Figure 2. Probabilistic time series for the simulation of soil moisture with behavioural 3

(NSE>0, bias<±10% soil moisture and R²>0.3) CMF-PMF model runs on site 1 for three soil 4

depths. Inserts: The likelihood functions quantify the median of the prediction range. 5

6

Fig. 2. Probabilistic time series for the simulation of soil moisture with behavioural (NSE> 0, bias< ±10 % soil moisture andR2 > 0.3)CMF–PMF model runs at site 1 for three soil depths. Inserts: the likelihood functions quantify the median of the prediction range.

days and lower during moderate moisture conditions. Butthe GLUE method per se has the tendency to overestimateuncertainty during low and high simulation events (Vrugt etal., 2008). In the soil depth from 0.3 to 0.6, as well as inthe soil depth from 0.6 to 0.9 m, we have a constant uncer-tainty in the prediction of around 5 %. The median of thebehavioural model run in the upper soil layer has an NSE of0.57, a bias of 2 % soil moisture and anR2 of 0.84. Modelperformance criteria in the soil depths below are of similarquality, with poorer performance forR2 values but improvedbiases (Fig. 2).

Compared with other studies, the median output forsoil moisture after calibration was on the same qualitylevel as previously reported findings. For example, Chris-tiaens and Feyen (2002) published results of the GLUEmethod used for the MIKE SHE model with an NSE closeto zero. Jiménez-Martínez et al. (2009), who simulated thesoil moisture under melons growing in southeastern Spain,found a van Genuchten parameter set for Hydrus-1D modelresulting in a highR2 of 0.9 and 0.029 % RMSE for soilmoisture. Scharnagl et al. (2011) found a RMSE of 0.009water content and NSE of 0.87 for their Hydrus-1D modelsetup at the site of Selhausen, near Jülich, Germany. Theiruncertainty bounds for soil moisture varied around 3 %, withhigher uncertainty during dry and wet situations, which isconsistent with our findings. We obtained similar model effi-ciencies as the best-performing model CERES (NSE= 0.66)in predicting the soil water content over all soil depths as a

study of Kersebaum et al. (2007) at the same study site 1 inMüncheberg.

Despite those results, the prediction uncertainty could befurther reduced by using more model runs and a stricter set-ting of threshold likelihood function. However, single modelrun time and the number of model runs had already pushedthe overall computer run time of the uncertainty estimationprovided here to three months. An efficient way might be theuse of the DREAM algorithm that is able to solve complexposteriori probability density functions for a large number ofparameters. This algorithm could reduce model runs and theuncertainty of the posteriori distribution (Vrugt et al., 2008).The GLUE method was chosen because of its easy imple-mentation and the possibility of parallelization.

Nevertheless, the simulated parameter uncertainty can alsodepend on the chosen likelihood function and is not indepen-dent of errors in measured data (Mo and Beven, 2004). Thus,instead of attributing remaining model predictive uncertaintyto the coupled CMF–PMF model structure itself, we shouldbe aware that there are other sources of global uncertaintythat impact on the overall model performance.

3.2.2 Plant growth

Results for the root, stem and leaves as well as storage drymatter are given in Fig. 3. This distribution shows good re-sults for the root dry-matter simulation. All observed valuesfall within the 50 % probability range. A high NSE of 0.94andR2 of 0.98 along with a very low bias of−58.2 kg ha−1



28

1

2

Figure 3. Probabilistic time series for the simulation of plant dry matters with behavioural 3

(NSE>0, bias<±500 kg ha-1

plant dry matter and R²>0.3) CMF-PMF model runs on site 1. 4

Inserts: The likelihood functions quantify the median of the prediction range. Note differences 5

in scale of y-axis. 6

7

Fig. 3.Probabilistic time series for the simulation of plant dry matter with behavioural (NSE> 0, bias< ±500 kg ha−1 plant dry matter andR2 > 0.3) CMF–PMF model runs at site 1. Inserts: the likelihood functions quantify the median of the prediction range. Note differences inthe scale of they axis.

indicate acceptable model performance. The median of thestem and leaves dry-matter simulation quality is lower thanfor the other simulated outputs with NSE of 0.79, bias of221.7 kg ha−1 and R2 of 0.87. Looking at the uncertaintyboundaries, we can locate a relatively large uncertainty start-ing especially from July onwards and a somewhat lower un-certainty during the first half of June, without matching theobserved value on 14 June. The observed value on this day iseven higher than the next observation on 26 July, which may,however, occur in reality owing to decaying leaves (senes-cence). In the current model version of PMF the model can-not represent a reduction of biomass during the growing sea-son due to leave senescence. In this sense, GLUE even fa-cilitates the investigation of the model structure and identifi-cation of clear model limitations. The uncertainty of the pre-diction is constant around 500 kg ha−1 for the root dry matter,while the stem and leaves as well as the storage dry matterhave a mean uncertainty of around 2000 kg ha−1. The storagedry-matter simulation fits the measured data within the 50 %probability boundary.

In comparison to previously reported studies, we obtainedacceptable results for the prediction of plant dry matter. Forexample, Jégo et al. (2010) found an RMSE of 1000 kg ha−1

for spring wheat biomass simulation using the STICS model.Results are also excellent when compared to the model inter-comparison study of plant growth models that was realizedfor the same forcing and evaluation data set by Kersebaumet al. (2007). Eight models were applied, resulting in RM-

SEs from 773 kg ha−1 to 3329 kg ha−1 and NSE spanningfrom 0.19 to 0.96 in simulation of above-ground biomass forsite 1. The best-performing model was AGROSIM (note thatthis was the worst-performing model in the intercomparisonsproject by Kersebaum et al., 2007, who looked at soil mois-ture prediction), while the CANDY model returned the worstresults.

3.3 Cross comparison of sites and parameter sets

We obtained reliable soil moisture and plant dry-matter out-puts for three experimental sites in Müncheberg with ourcoupled CMF–PMF model. But the observed data set is rel-atively small, as is mostly the case in measured plant dataanalyses. Therefore it is even more essential to evaluatemodel performance across different field sites. We thereforechose to apply a cross-validation method in which parame-ter settings for one site were tested on another site and viceversa in order to investigate the general model and parametertransferability (Pathak et al., 2012). This procedure has be-come an established method in dealing with small data setsin the course of model parameterization, calibration and val-idation (Nassif et al., 2012).

A comparison of the range of the behavioural parametersets of the three sites is shown in Fig. 4. We can see small andsimilar ranges for the most-constrained parameters – RUE,tbase, alpha andn – over all sites. In this case, site 1 showsthe widest range for the constrained parametersksat, alpha,



29

1

2

Figure 4. Range of behavioural parameter sets considering all three threshold criteria of the 3

CMF-PMF model for the three sites in Muencheberg. Results are shown for the same set of 4

model input parameters as in Fig. 1. 5

6

Fig. 4.Range of behavioural parameter sets considering all three threshold criteria of the CMF–PMF model for the three sites in Müncheberg.Results are shown for the same set of model input parameters as in Fig. 1.

30

1

2

Figure 5. Cross-validation of soil moisture prediction with uncertainty boundaries. Grey 3

shaded sites are calibrated. Black dots are observed values, red dashed line is the median of 4

the behavioural boundary condition (NSE>0, bias<±10% soil moisture and R²>0.3). The 5

yellow area is the 95% probability range of the simulation, the orange area the 50% 6

probability range. 7

8

Fig. 5. Cross validation of soil moisture prediction with uncertainty boundaries. Grey-shaded sites are calibrated. Black dots are observedvalues, and the red dashed line is the median of the behavioural boundary condition (NSE> 0, bias< ±10 % soil moisture andR2 > 0.3).The yellow area is the 95 % probability range of the simulation, and the orange area the 50 % probability range.

tbaseand root growth. Medians shown as red lines in the boxplots indicate optimal parameter settings for the constrainedparameters. They are located more or less at the same posi-tion for the four constrained parameters, while this positionvaries throughout the sites for the other parameters. We con-clude that in further applications of CMF–PMF, ranges forthe constrained parameters as given in Table 1 can be sub-stantially reduced to obtain improved model runs. One couldalso consider fixing the parameter to the median and exclud-ing them from further calibration.

We employed a cross validation with each of the be-havioural parameter sets we obtained for one site on the re-maining two sites. As examples we show results of the crossvalidation for soil moisture 0.3–0.6 m [%] (Fig. 5) as well asfor stem and leaf (Fig. 6). Transferability of model parame-ter sets worked well for soil moisture. In comparison to theother sites, we found the smallest uncertainty ranges to be atsite 3. While site 1 has a mean uncertainty around 10 %, site3 has only 5 %. Nevertheless, parameter sets found for site 1worked well for the other sites. The NSE dropped from a high



31

1

2

Figure 6. Cross-validation of stem and leaves prediction with uncertainty boundaries. Grey 3

shaded sites are calibrated. Black dots are observed values, red dashed line is the median of 4

the behavioural boundary condition (NSE>0, bias<±500 kg ha-1

plant dry matter and R²>0.3). 5

The yellow area is the 95% probability range of the simulation, the orange area the 50% 6

probability range. 7

Fig. 6.Cross validation of stem and leaves prediction with uncertainty boundaries. Grey-shaded sites are calibrated. Black dots are observedvalues, and the red dashed line is the median of the behavioural boundary condition (NSE> 0, bias< ±500 kg ha−1 plant dry matter andR2 > 0.3). The yellow area is the 95 % probability range of the simulation, and the orange area the 50 % probability range.

level of 0.48 for site 1 to NSE= 0.31 for site 2 and an NSEof 0.37 for site 3. The same cross validation at the other sitesresulted in a small but constant range of NSE between 0.23and 0.37. The bias remains the same across all sites rangingbetween very low soil moisture of−0.6 and−1.0 %.

Cross validation for stem and leaves output of the CMF–PMF model worked well for sites 1 and 3, but less well forsite 2 (Fig. 6). This is most likely related to the similar ob-served values for the stem and leaves dry matter at the inten-sively managed site (site 1) and the extensively managed site(site 3), while the stem and leaves dry matter at the organi-cally managed site (site 2) is significantly lower. Uncertaintyboundaries for the organic site growths during the simulationperiod are up to 2000 kg ha−1, while the uncertainty for theother sites increases up to 3000 kg ha−1, with a very low un-certainty around the 14 June 1994. We found similar NSEs of0.79 for site 1 and 0.74 for site 2 and 3. Validated on the othersites, parameter settings for site 1 resulted in an acceptableNSE of 0.35 for site 2 and a good NSE of 0.88 for the ex-tensive site (site 3).R2 remains on the same level through alltested sites, between 0.79 and 0.91, indicating that the gen-eral dynamics of crop growth were captured for all sites. The

variation of performance criteria in the cross-validation ex-periment (i.e. stableR2 across all sites versus a drop of NSEand an increase in bias from one site to another) highlight theimportance of using a set of different likelihood functions.

One source of uncertainty in the prediction quality of thecoupled model with regard to dry-matter production is to beseen in our disregarding of further field management strate-gies. Even though fertilization is considered in the simulationof crop growth in the current model setup of CMF–PMF, weneglected other agricultural management options that signif-icantly influence biomass production, e.g. pesticide applica-tion in conventional field management or weeding in organicfarming. Accordingly, the water balance of the soil has beensimulated without a nutrient balance. For this reason, onlywater stress restricts crop growth. Fertilizer demand does notconstrain plant growth in our model setup, as fertilizer is pro-vided as unlimited. In general, PMF is capable of accountingfor active and passive nitrogen uptake. Selection of cultivarsalso has a significant impact on yields, which is also not con-sidered in PMF. The reason for this is simple: PMF does nothave a management tool. Site 1 was managed with a high-level intensive strategy, site 2 an organic one and site 3 an



extensive one. The winter wheat cultivar was also adapted tothese strategies with elite winter wheat ’Bussard’ for site 1,infrequently used ’Ramiro’ for site 2 and elite winter wheat’Greif’ for site 3. While these differences in management andcultivar lead to a high variability in plant matter productionacross sites (Fig. 6), they does not impact soil moisture con-ditions to a similar degree (Fig. 5). Consequently, to applyPMF in the sense of a full crop growth model for agriculturalapplication, a management module is required that considerstypical management strategies in agriculture. Instead of fullinclusion of this management tool in the PMF model itself,we promote following the idea of the framework strategy ofPMF as well as CMF and apply an external farm manage-ment model (Aurbacher et al., 2013; Windhorst et al., 2012).

4 Conclusions

In keeping in line with standards for the development andevaluation of environmental models, our implementation offour consecutive steps in implementing the GLUE method(sect. 2.2.2) for the validation of our coupled model is consis-tent with the postulations of Jakeman et al. (2006). Throughthe investigation of the parameter uncertainty, the CMF–PMF model performance was found to crucially depend onthe parameter values forn (CMF) and RUE (PMF). Theiruncertainty boundaries could be reduced by 60 and 77 %,respectively, through the GLUE analyses. Other parameters– including k, emergence, stem elongation and anthesis –showed only a minor influence on the model outputs. Theperformance of our CMF–PMF model setup was found to bebetter than some previously tested models given that modelperformance was good for soil moisture and plant dry mat-ter at the same site. Overall, approximately 90 % of theobserved soil moisture data were within the predicted un-certainty bounds that were determined through the GLUEmethod. The model performances for simulating observedplant dry matter were found to be in an uncertainty rangefrom 500 to 2000 kg ha−1, with only one measured valuemissed. The posteriori parameter settings found can be usedfor a more efficient calibration of the CMF–PMF model infuture case studies.

The cross validation at different sites showed only slightreductions of the likelihood functions. From this, we con-clude that the model is transferable in space, at least undersimilar soil and weather conditions. Next steps should in-clude a model test over several growing periods for whichother crops need to be covered by PMF to be able to simulatecrop rotation patterns.

Structural model uncertainty was identified with regard tothe need of including agricultural management options andthe missing capability of representing senescence in PMF.While the latter should be improved by considering processesreflecting senescence within PMF, we propose coupling of an

agricultural management model and CMF–PMF, an essentialstep if PMF is to be used as a full crop growth model.

Acknowledgements.We would like to thank K. C. Kersebaum andthe Leibniz Centre for Agricultural Landscape Research (ZALF)for providing their comprehensive data set from the Münchebergfield sites. The financial support provided by the Gesellschaft fürBoden- und Gewässerschutz e.V. as well as the DFG (BR2238/13-1) is gratefully acknowledged.

Edited by: G. Wohlfahrt

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